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Συντελεστής Lorentz
Συντελεστής Lorentz Lorentz factor Σχετικιστική Πεδιακή Θεωρία Σχετικιστική Κλασσική Μηχανική Σχετικιστική Κβαντική Μηχανική Ειδική Σχετικότητα Γενική Σχετικότητα Χωροχρόνος Χώρος Minkowski Σχετικιστικοί Φυσικοί Γης]] thumb|300px|[[Συντελεστής Lorentz.]] thumb|300px|[[Συντελεστής Lorentz.]] - Ένα Γεωμετρικό Μέγεθος. Ετυμολογία Η ονομασία "Lorentz factor" σχετίζεται ετυμολογικά με το όνομα "Lorentz". Ορισμός The Lorentz factor is defined as:Dynamics and Relativity, J.R. Forshaw, A.G. Smith, Wiley, 2009, ISBN 978-0-470-01460-8 : \gamma = \frac{1}{\sqrt{1 - v^2/c^2}} = \frac{1}{\sqrt{1 - \beta^2}} = \frac{\mathrm{d}t}{\mathrm{d}\tau} where: * v'' is the relative velocity between inertial reference frames, * β is the ratio of ''v to the speed of light c''. * ''τ is the proper time for an observer (measuring time intervals in the observer's own frame), * c'' is the ''speed of light. Occurrence Following is a list of formulae from Special relativity which use γ'' as a shorthand: * The 'Lorentz transformation:' The simplest case is a boost in the ''x-direction (more general forms including arbitrary directions and rotations not listed here), which describes how spacetime coordinates change from one inertial frame using coordinates (x'', ''y, z'', ''t) to another (x' '', ''y' '', ''z' '', ''t' '') with relative velocity ''v: :: t' = \gamma \left( t - \frac{vx}{c^2} \right ) :: x' = \gamma \left( x - vt \right ) Corollaries of the above transformations are the results: * Time dilation: The time (∆''t' ) between two ticks as measured in the frame in which the clock is moving, is longer than the time (∆''t) between these ticks as measured in the rest frame of the clock: :: \Delta t' = \gamma \Delta t. \, * Length contraction: The length (∆''x' ) of an object as measured in the frame in which it is moving, is shorter than its length (∆''x) in its own rest frame: :: \Delta x' = \Delta x/\gamma. \,\! Applying conservation of momentum and energy leads to these results: * Relativistic mass: The mass of an object m'' in motion is dependent on \gamma and the rest mass ''m''0: :: m = \gamma m_0. \, * 'Relativistic momentum:' The relativistic momentum relation takes the same form as for classical momentum, but using the above relativistic mass: :: \vec p = m \vec v = \gamma m_0 \vec v. \, Προσέγγιση Ισχύει για την Square root: : \sqrt{1+x} = \sum_{n=0}^\infty \frac{(-1)^n(2n)!}{(1-2n)(n!)^2(4^n)}x^n = 1 + \tfrac{1}{2} x - \tfrac{1}{8} x^2 + \tfrac{1}{16} x^3 - \tfrac{5}{128} x^4 + \dots\quad\text{ for }|x|\le1 οπότε: : (1-x^2)^{1/2}=1-\frac{x^2}2-\frac{x^4}8-\frac{x^6}{16}\cdots : \cosh x = 1 + \frac {x^2} {2!} + \frac {x^4} {4!} + \frac {x^6} {6!} + \cdots = \sum_{n=0}^\infty \frac{x^{2n}}{(2n)!} αλλά: : c^2 = (v^2 + c^2 - v^2) = (v^2 + c^2 (1 - v^2/c^2)) Οπότε: : \gamma = \left+ \frac{1}{2} \left(\frac{v}{c}\right)^2 + \frac{3}{8} \left(\frac{v}{c}\right)^4 + \frac{5}{16} \left(\frac{v}{c}\right)^6 + \ldots \right. For speeds much smaller than the speed of light, higher-order terms in this expression get smaller and smaller because ''v/''c'' is small. For low speeds we can ignore all but the first two terms: : \gamma c^2 \approx c^2 + \frac{1}{2} v^2 . Αριθμητική Τιμή Οι αριθμητικές τιμές που λαμβάνει ο συντελεστής γ ως συνάρτηση της ταχήτητας) είναι: The middle column shows the corresponding Lorentz factor, the final is the reciprocal. Υποσημειώσεις Εσωτερική Αρθρογραφία * τανυστής Einstein * ενέργεια * ορμή * συντελεστής Βιβλιογραφία * * Ιστογραφία *Ομώνυμο άρθρο στην Βικιπαίδεια *Ομώνυμο άρθρο στην Livepedia *Ομώνυμο άρθρο στην Astronomia *[ ] *[ ] Category: Σχετικιστική Φυσική